Step by step
The question
Question: find f(x), if f′(x) = 3x² + 2 and f(1) = 4.
f′(x) = 3x2 + 2
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
f(x) = 3 × x33 + 2x + C
Now we have f(x) = x³ + 2x + C.
= x3 + 2x + C
Every antiderivative has + C.
The point will tell us what C is.
+ C: we do not know it yet
2 Find C
Put in x = 1.
f(1) = 13 + 2(1) + C
Work it out: 3 + C.
= 3 + C
It must equal 4.
3 + C = 4
Take away 3 on both sides: C = 1.
C = 4 − 3 = 1
Careful!
C is not just the f value.
The other terms at x = 1 are not 0 here.
Careful: not C = 4 ✗
3 Write the function
So f(x) = x³ + 2x + 1.
f(x) = x3 + 2x + 1
4 Check
Let's check: its derivative is 3x² + 2, the slope we were given.
f′(x) = 3x2 + 2
And at x = 1 it gives 4.
Right!
x = 1: 3 + 1 = 4
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer
f(x) = x3 + 2x + 1